jmc

To get the equation of the asymptotes, we replace the $1$ on the right-hand side with $0,$ giving the equation $$\frac{y^2}{9}-\frac{x^2}{4}=0.$$(Notice that there are no points $(x,y)$ which satisfy both this equation and the given equation, so as expected, the hyperbola never intersects its asymptotes.) This is equivalent to $\frac{y^2}{9}=\frac{x^2}{4},$ or $\frac{y}{3}=\pm \frac{x}{2}.$ Thus, $y=\pm \frac{3}{2}x,$ so $m=\boxed{\frac{3}{2}}.$